The US has the same caveats, it's just that the limitations of free speech are decided in case law instead of specifically codified.
> For the prevention of disorder of crime
There are all sorts of types of speech that are criminal in the US because they are either a crime, or encourage/enable crime in some way.
> for the protection of health and morals
There are plenty of examples in the US of speech being limited for health reasons.
Protection of morals is a bit more of a gray area, but certainly there are historical examples of the US limiting speech for reasons that amount to morality.
> for the protection of the reputation or rights of others
Again, libel is a crime in the US.